Explore topic-wise InterviewSolutions in Current Affairs.

Answer is (a) 1/√35,3/√35,5/√35

(A) A, B and C are collinear

Area under normal curve to the right of z = 1.3

= Area from 0 to ∞ – Area from 0 to 1.3

= 0.5 – 0.4032 = 0.0968

Answer is (c) ll₁ + mn₁ + nn₁ = 0

A ‘defect’ is a quality characteristic which does not conform to specifications. 

The presence of one or more defects in a product is ‘defective.

(a) Range is ( -∞ , ∞ )

(b) Mean = 0 (mean = median = mode = 0).

Fisher’s Index Number is called as ideal because, it is free from bias in use of weights ie. it takes both current and base year quantities as weights. 

• It is based on geometric mean which is considered as best average 

• It satisfies both time reversal tests and factor reversal tests

Answer is (b) (x - 2)/3 = (y + 1)/-1 = (z - 3)/2

They are used in computation of actuarial of premium, bonus etc, of policies by Insurance Agencies. 

OR 

• Life Tables are used in research activities in Biology, Medicine, Pharmacology, Demography, Psychology, Sociology etc. 

• They are used to study population growth and forecast the size and sex distribution of the Population. 

OR 

• Life Tables give Mortality and Survival ratios at different ages. 

• Life tables give the life expectancy at different age. 

OR 

• These are useful in public administration, heath care, planning and population control

(a) To test whether a population has given variance. 

(b) To test ‘goodness of fit’ of theoretical frequencies to a observed frequencies.

Answer is (b) 3l = 2m = n

Replacement’s carried out when 

(a) Machine and their spare parts deteriorates with time. 

(b) The maintenance cost increases and efficiency reduces.

Chronological arrangement of statistical data according to time is called ‘Time series’.

An Inventory is a physical stock of goods, which is held for the purpose of future production or sales.

Current year quantity (q₁).

Answer is (d) 2

• As the equipment grow older, its maintenance cost would be costlier. 

• Production cost would be lesser in new equipments.

Answer is (b) 1/2

Answer is (b) P(AB') = P(A) P(B')

Answer is (c) 10/3

Here no. positive allocations 

= 4; m + n – 1 = No. of rows + No. of columns -1 = 3 + 3 -1 = 5 

Here no. of allocations < (m + n – 1) the solution is degenerate / not a non-degenerate.

• A defect is a quality characteristic, which does not conform to specifications. 

• The presence of one or more defects in a product/article is a defective.

Answer is (a) 40

Answer is (d) 2tan^-1 x

z²+pz+q=0

z₁+z₂ = p, z₁z₂=q

If z₁,z₂,z₃ are the vertices of an equilateral triangle then z₁²+z₂²+z₃² = z₁z₂+z₂z₃+z₃z₁

If z₃ is origin then z₁²+z₂²=z₁z₂

=> (z₁+z₂)²=3z₁z₂

=> p² = 3q

Let , a/R, a, aR be the roots of x³ – px² + qx – r = 0

Then product of the roots = a/R .a. aR = a³ = r ⇒ a = r^1/3

∵ a is a root of x³ – px² + qx – r = 0

⇒ a³ – pa² + qa – r = 0

But a = r^1/3

⇒ (r^1/3)³ – p(r^1/3)² + q(r^1/3) – r = 0

⇒ r – p. r^2/3 + q. r^1/3 – r = 0 ⇒ p. r^2/3 = q r^1/3

By cubing on both sides

⇒ p³r² = q³r

⇒ p³r = q³ is the required condition.

Given Z = (1 + 2i)/(1 - (1 - i)²)

= (1 + 2i)/(1 - (1 - 2i + i²)) = (1 + 2i)/(1 + 2i) = 1

= cos0° + isin0°

∴ Arg(Z) = 0°

(–16) = 16(–1) = 16(cosπ + isinπ)

= 2⁴(cos(2k + 1)π + isin(2k + 1)π], k ∈ z

(–16)^1/4 = (2⁴)^1/4[cos(2k + 1)π + i sin (2k + 1)π]^1/4

= 2cos(2k + 1)π/4, k = 0, 1, 2, 3

The value of (–16)^1/4 is

2cos(2k + 1)π/4, k = 0, 1, 2, 3

Let Z = x + iy. Then |Z| = 2.

If and only if |x + iy| = 2.

If and only if √(x² + y²) = 2.

If and only if x² + y² = 4

x² + y² = 4 represents a circle with centre at (0, 0) and radius 2!

∴ The locus of |z| = 2 is the circle x² + y² = 4. 

(B) x logx – x + c 

Correct option :

(A) [0,1] 

correct option :

(C) 0.8 

Correct option:

(D) 4/5

(A) x dy/dx = y

(C) y = x² + kx 

Correct option:

(C) 2

correct option:

(A) 2

Answer is (b) log(1 + y) = x + (x²/2) + k

Answer is (d) 4/5

\(\frac{x}{x+2}\leq\frac1{|x|}\) 

⇒ \(\frac{x|x|}{x+2}\leq1\) (Multiplying both sides by positive value |x|)

I₆ x = -3, then \(\frac{x|x|}{x+2}=\frac{-3\times3}{-3+2}=\frac{-9}{-1}=9>1\) (Not satisfies)

(\(\therefore\) I₆ x < -2, then x|x| < 0 & x + 2 < 0  ⇒ \(\frac{x|x|}{x+2}>1\))

x = -2 is not possible

x = -1 then \(\frac{x|x|}{x+2} = \frac{-1\times1}{-1+2}=\frac{-1}1=-1<1\) (Satisfies)

x = 0 then \(\frac{x|x|}{x+2} = 0<1\) (Satisfies)

x = 1 then \(\frac{x|x|}{x+2} = \frac13<1\) (Satisfies)

x = 2 then \(\frac{x|x|}{x+2} = \frac{2\times2}{2+2}=\frac44=1\) (Satisfies)

I₆ x > 2 then x|x| = x² > x + 2

⇒ \(\frac{x|x|}{x+2} >1\) (Not satisfies)

\(\therefore\) Number of integral values which satisfies given inequality are 4.

LHS = {cosx / sinx - sinx / cosx} / {cosx / sinx + sinx / cosx}

LHS = {cos²x - sin²x / sinxcosx} / {cos²x + sin²x / sinxcosx}

LHS = cos²x - sin²x = cos2x... (cos²x + sin²x = 1)

LHS = cos2x = sinx = RHS. 

Hence, proved. 

\(\frac{cotx-tanx}{cotx+tanx}=\cfrac{\frac{cosx}{sin x}-\frac{sin x}{cos x}}{\frac{cosx}{sinx}+\frac{sinx}{cosx}}\) 

\(=\cfrac{\frac{cos^2x-sin^2x}{sinxcosx}}{\frac{cos^2x+sin^2x}{sinxcosx}}\) = \(\frac{cos^2x-sin^2x}{sin^2x+cos^2x}\)

\(= cos^2x-sin^2x\) (\(\because\) sin²x + cos²x = 1)

 = cos 2x (\(\because\) cos 2x = cos²x - sin²x)

\(\therefore\) \(\frac{cotx-tanx}{cotx+tanx}=cos2x\neq sinx\)

The mass of a single tungsten atom is 3.05 × 10^-22 grams.

Dot Product Properties:

• Dot product of two vectors is commutative i.e. a.b = b.a = ab cos θ.
• If a.b = 0 then it can be clearly seen that either b or a is zero or cos θ = 0 ⇒θ = π/2. It suggests that either of the vectors is zero or they are perpendicular to each other.
• Also we know that using scalar product of vectors (pa).(qb)=(pb).(qa)=pq a.b
• The dot product of a vector to itself is the magnitude squared of the vector i.e. a.a = a.a  cos 0 = a²
• The dot product follows the distributive law also i.e. a.(b + c) = a.b + a.c

Properties of the Cross Product:

• The length of the cross product of two vectors is \(|\vec a\times\vec b|=|a||b|sin\theta\)
• The length of the cross product of two vectors is equal to the area of the parallelogram determined by the two vectors.
• Anticommutativity: \(\vec a\times\vec b=-\vec b\times\vec a\)
• The volume of the parallelepiped determined by the vectors a, b, and c is the magnitude of their scalar triple product

We know that 

T \(\propto\) m^a l^a g^c

T = K m^a l^a g^c----(1)

Putting the dimension each parameter both side

[T] = [M]^a [L]^b[LT^-2]^c

[T] = [M]^a[L^b+c][T^-2c]

comparing both sides power

a = 0

b + c = 0

-2c = 1

c = -1/2

these value put in equation (1) we get

T = K m⁰ l^1/2 g^-1/2

T = K\(\sqrt{\frac lg}\) \(\because k = 2\pi\)

T = 2\(\pi\sqrt{\frac lg}\)